Two-Grid Finite Element Discretization Schemes Based on Shifted-Inverse Power Method for Elliptic Eigenvalue Problems

Abstract

This paper discusses highly efficient discretization schemes for solving self-adjoint elliptic differential operator eigenvalue problems. Several new two-grid discretization schemes, including the conforming and nonconforming finite element schemes, are proposed by combining the finite element method with the shifted-inverse power method for matrix eigenvalue problems. With these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$ and the solution of a linear algebraic system on the fine grid $\pi_{h}$. Theoretical analysis shows that the schemes have a high efficiency. For instance, the resulting solution can maintain an asymptotically optimal accuracy by using the conforming linear element or the nonconforming Crouzeix–Raviart element by taking $H=O(\sqrt[4]{h})$. Numerical experiments are presented to support the theoretical analysis. In addition, this paper establishes multigrid discretization schemes and proves their efficiency.